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Groupoids assigned to relational systems. (English) Zbl 1274.08002

Summary: By a relational system we mean a couple \((A,R)\) where \(A\) is a set and \(R\) is a binary relation on \(A\), i.e., \(R\subseteq A\times A\). To every directed relational system \(\mathcal {A}=(A,R)\) we assign a groupoid \({\mathcal G}({\mathcal A})=(A,\cdot )\) on the same base set where \(xy=y\) if and only if \((x,y)\in R\). We characterize basic properties of \(R\) by means of identities satisfied by \({\mathcal G}({\mathcal A})\) and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.

MSC:

08A02 Relational systems, laws of composition
20N02 Sets with a single binary operation (groupoids)
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